After listing contract expires can I list immediately with a new broker?

The answer to the subject line is yes. Most brokers work for the seller quite often, shoot an email to everyone you know and ask for reccomendations as to good brokers, then interview a few. Given that I am gathering from your question that you have had the property on the market, you really want these brokers to give you insight as to why it has not sold, and what needs to be done to sell it.

The reason your apartment did not Sell is probably not the Broker. It is likely because either (a) it is priced 15% or more above actual market prices so buyers are turned off by the price or (b) there is a major problem with your apartment like it is below street level, needs major renovation, extremely high maintenance, undesireable location. You should be able to price your apartment correctly based on reviewing actual recorded sales data from 2007 in StreetEasy. If the kitchen and baths can be modernized, do so within a reasonable budget. If there is a problem with light, air, land (location), legal structure (bad co-op), then there is no way to fix it. Either way, if the broker advertised the place, held open houses, and responded to qualified inquiries, they did all they can do. Brokers have no special powers. A broker working for one of the major NYC brokerage firms is just holding your hand and taking 6% of the value of your real estate. They are not adding any value to the purchase price; they are just doing a lot of administrative work that is annoying and inconvenient for you to do yourself.

#3, shouldn't the broker offer some guidance on how to price? Also, it may be they haven't done very much to market the place. A new broker might be more helpful on both these fronts.

#4, this is #3. You are absolutely right that the Broker should be the driving force in setting a realistic -- if slightly optimistic -- initial selling price that is based on actual market data (comparable recorded sales). Many brokers have absolutely no knowledge or training in business, economics, or finance, and are completely clueless about pricing an apartment appropriately. Given that Manhattan real estate has been in a raging bull market since 1994 (prices have increased 250%), most brokers these days win listings by setting grossly inflated price expectations for Sellers. When that price does not stick, the Seller ends up in the situation like OP is in. Sometimes they get lucky and a Greater Fool shows up to buy it. Sometimes they are right like when a European (Euros) or Saudi (petrodollars) shows up. A ground floor 350 square foot co-op studio with a 200 square foot private backyard on the Upper West Side sold in our building for $625,000 to the daughter of a Saudi oilman. Some people will pay anything. Good luck, OP!